Step by step
The question
Question: find the distance between (1, 1, 1) and (3, 3, 2).
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (1, 1, 1).
Point B: (3, 3, 2).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from 1 to 3.
3 minus 1 is 2.
Along y: y goes from 1 to 3.
3 minus 1 is 2.
Up z: z goes from 1 to 2.
2 minus 1 is 1.
3 Square and add
On the flat, the x and y moves make a right triangle.
Its long side is this green line.
The green line and the move up z make a second right triangle.
Its long side is AB.
So Pythagoras works twice.
Square the three differences, add them, then take the square root.
2 squared is 4.
2 squared is 4.
1 squared is 1.
4 Take the square root
Add: 4 plus 4 plus 1 is 9.
Careful!
9 is d squared.
We still take the square root.
3 times 3 is 9.
So d = 3.
5 Check
Let's check: square the distance.
We get 9, the same as 4 plus 4 plus 1.
Right!
Going from B to A instead only changes the signs.
The squares are the same.
Right!
Let's look back.
First: plot the points.
Then: the differences.
Then: square and add.
Then: take the square root.
Last: check.